Abstract
The present paper introduces a new approach to simulate any stationary multivariate Gaussian random field whose cross-covariances are predefined continuous and integrable functions. Such a field is given by convolution of a vector of univariate random fields and a functional matrix which is derived by Cholesky decomposition of the Fourier transform of the predefined cross-covariance matrix. In contrast to common methods, no restrictive model for the cross-covariance is needed. It is stationary and can also be reduced to the isotropic case. The computational effort is very low since fast Fourier transform can be used for simulation. As will be shown the algorithm is computationally faster than a recently published spectral turning bands model. The applicability is demonstrated using a common numerical example with varied spatial correlation structure. The model was developed to support simulation algorithms for mineral microstructures in geoscience.
Highlights
The theory of uni- and multivariate random fields has been used extensively in a broad range of science and engineering disciplines such as meteorology, astrophysics and geosciences ([1] [2]) over the last decades
The present paper introduces a new approach to simulate any stationary multivariate Gaussian random field whose cross-covariances are predefined continuous and integrable functions
Such a field is given by convolution of a vector of univariate random fields and a functional matrix which is derived by Cholesky decomposition of the Fourier transform of the predefined cross-covariance matrix
Summary
The theory of uni- and multivariate random fields has been used extensively in a broad range of science and engineering disciplines such as meteorology, astrophysics and geosciences ([1] [2]) over the last decades. The spectrum convolution approach proposed in this paper creates exact stationary Gaussian random fields Such a multivariate field with values in N is given by convolution of a N dimensional vector of univariate Gaussian random fields and a functional N × N matrix. Note that this model is completely different from the Cholesky decomposition approach ([15] [16]) and requires only decomposition of. Section two introduces notation and provides the theoretical frameworks and results for multivariate random fields as well as a closer look at two other approaches This is followed by a description of the new.
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